Hypercyclic operators that commute with the Bergman backward shift

The backward shift B on the Bergman space of the unit disc is known to be h ypercyclic (meaning: it has a dense orbit). Here we ask: "Which operators t hat commute with B inherit its hypercyclicity?" We show that the problem re duces to the study of operators of the form phi(B) where phi is a holomorph ic self-map of the unit disc that multiplies the Dirichlet space into itsel f, and that the question of hypercyclicity for such an operator depends on how freely phi(z) is allowed to approach the unit circle as \z\ --> 1-.